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Sunday, 8 September 2013

Generalized Hadamard Gates

Consider the action of the Hadamard gate \(H\) when acting on the basis states \(\left|0\right>\) and \(\left|1\right>\), and observe that this can be expressed as
\[\begin{array}{r l}
H\left|x\right> &= \frac{1}{\sqrt{2}}\big(\left|0\right>+(-1)^{x}\left|1\right>\big) \\
& =\frac{1}{\sqrt{2}}\displaystyle\sum\limits_{y\in\{0,1\}}(-1)^{x\cdot y}\left|y\right>.
\end{array}\]
Denote some bitstring \(\mathbf{x}\in\{0,1\}^n\) as \(\mathbf{x}=x_1x_2\dots x_n\), where each \(x_i\in\{0,1\}\), and define the following operation on bitsrings
\[\mathbf{x}\cdot \mathbf{y}=(x_1y_1+x_2y_2+\dots +x_ny_n)mod2.\]
The tensor product of Hadamard gate \(H\) with itself \(n\) times creates a gate similar to \(H\), denoted by \(H^{\otimes n}\), which operates on \(n\) qubits. Its action is the same as applying the single qubit Hadamard transform \(H\) to each of the \(n\) qubits of the register. This gate sends the \(n\) qubit register \(\left|\mathbf{x}\right>=\left|x_1x_2\dots x_3\right>=\left|x_1\right>\left|x_2\right>\dots\left|x_n\right>\) to the state
\[\begin{array}{r l}
H^{\otimes n}\left|\mathbf{x}\right>&=H^{\otimes n}\left|x_1x_2\dots x_3\right> \\ & =H\left|x_1\right>H\left|x_2\right>\dots H\left|x_n\right>\\
& = \big(\frac{\left|0\right>+(-1)^{x_1}\left|1\right>}{\sqrt{2}}\big)\big(\frac{\left|0\right>+(-1)^{x_2}\left|1\right>}{\sqrt{2}} \big)\dots \big(\frac{\left|0\right>+(-1)^{x_n}\left|1\right>}{\sqrt{2}}\big) \\
& = \frac{1}{\sqrt{2^n}}\displaystyle\sum\limits_{y_1y_2\dots y_n\in\{0,1\}^n}(-1)^{x_1y_1+x_2y_2+\dots +x_ny_n}\left|y_1\right>\left|y_2\right>\dots\left|y_n\right> \\
& = \frac{1}{\sqrt{2^n}}\displaystyle\sum\limits_{\mathbf{y}\in\{0,1\}^n}(-1)^{\mathbf{x}\cdot \mathbf{y}}\left|\mathbf{y}\right>.
\end{array}\]
In the case where \(\left|\mathbf{0}\right>=\left|00\dots 0\right>=\left|0\right>\left|0\right>\dots\left|0\right>,\)
\[H^{\otimes n}\left|\mathbf{0}\right>= \frac{1}{\sqrt{2^n}}\displaystyle\sum\limits_{\mathbf{y}\in\{0,1\}^n}\left|\mathbf{y}\right>,\]
since \((-1)^{\mathbf{x}\cdot \mathbf{0}}=1\), which is an equally weighted superposition of all bases states in the Hilbert space of dimension \(2^n\).